# The minimal observable clade size of exchangeable coalescents

## TL;DR

This paper investigates the asymptotic behavior and moments of the minimal observable clade size in exchangeable coalescent models with mutations, providing insights relevant for genetic data analysis and model selection.

## Contribution

It introduces asymptotic results and recursive formulas for the moments of the minimal observable clade size in $ ext{Lambda}$-coalescents with mutations, a quantity previously unobservable in real data.

## Key findings

- Asymptotic behavior of $O_n$ as $n o
$ derived.
- Recursive formulas for all moments of $O_n$ established.
- $O_n$ provides an upper bound for the minimal clade size.

## Abstract

For $\Lambda$-$n$-coalescents with mutation, we analyse the size $O_n$ of the partition block of $i\in\{1,\ldots,n\}$ at the time where the first mutation appears on the tree that affects $i$ and is shared with any other $j\in\{1,\ldots,n\}$. We provide asymptotics of $O_n$ for $n\to\infty$ and a recursion for all moments of $O_n$ for finite $n$. This variable gives an upper bound for the minimal clade size [2], which is not observable in real data. In applications to genetics, it has been shown to be useful to lower classification errors in genealogical model selection [10].